Let us now describe the recursive ENO algorithm for determining the
optimal stencil for the smoothest reconstruction of 
. All the
information concerning the smoothness of can be
extracted from a table of divided differences of 
. Employing a standard
notation, see Atkinson (1988) [1], the 
divided difference of
can be defined recursively as by:
It can be shown that if the function is smooth in the
interval 
but is discontinous in 
,
then, for a 
small enough :
Hence, we can compare the relative smoothness of the function
in two intervals defined by an equal number of adjacent segments by
comparing their corresponding divided differences. This fact gives us a
powerful tool to select automatically the best stencil for the smoothest
interpolation. Since all the points of the interpolating stencil must be
contigous, to specify a stencil we need only one of the two extreme points,
usually the left-most one, and the number of points in the stencil. The
smoothest-possible stencil can be built in an iterative way applying
the following algorithm:
where the function is reconstructing, the first point in
the stencil will be 
. We can represent this choice
by indicating the relative index 
.
, where 
is the order of
the polynomial.